Spin system and Pauli exclusion principle

introduction

• the simplest example of quantum mechanical system
  
• quantization of the angular momentum
  
• measures as being some multiple of Planck's constant divided by 2pi
  

 

 

 

고전역학의 각운동량

 

• \(\mathbf{L}=\mathbf{r}\times \mathbf{p}\)

 

• A classical electron moving around a nucleus in a circular orbit
  
  • orbital angular momentum, \(L=m_evr\)
  • magnetic dipole moment, \(\mu= -evr/2\)
  • where e, m_e, v, and r are the electron´s charge, mass, velocity, and radius, respectively.
• A classical electron of homogeneous mass and charge density rotating about a symmetry axis
  
  • angular momentum, \(L=(3/5)m_eR^2\Omega\)
  • magnetic dipole moment, \(\mu= -(3/10)eR^2\Omega\), where R and \Omega are the electron´s classical radius and rotating frequency
• gyromagnetic ratio \(\gamma = \mu/L=-e/2m_e\)
  [/pages/7141159/attachments/4562863 I15-62-g20.jpg]
  
• pictures from Gyromagnetic Ratio and Anomalous Magnetic Moment

 

 

 

 

 

 

representation theory

• concept from the representation of  \(SU(2)\)
  

• half of highest weight is called the spin of the module
  
  • Casimir operator can also detect this number.
    
• spin \(1/2\) is the most important case since they are the matter particles
  
• this is why we have half-integral spin although those representation are integral highest weight representations.
  

 

 

 

 

sl(2)

• commutator
  \([E,F]=H\)
  \([H,E]=2E\)
  \([H,F]=-2F\)
  

 

 

spin particle statstics

• Bosons
  
  • photon
  • vector boson
  • Gluon
  • follows Bose-Einstein statistics
  • force-transmitting particles
• Fermions = spin- \(1/2\) particles
  
  • quarks and leptons
    
  • follows Fermi-Dirac statistics
    
  • matter particles
    

 

 

Dirac wave equation[[search?q=dirac&parent id=6452627|]]

• Dirac equation